Interior Point Algorithm

Delta Air Lines flies over 2,500 domestic flight legs every day, using about 450 aircraft from 10 different fleets. The fleet as-signment problem is to match aircraft to flight legs so that seats are fllled with paying passengers. Recent advances in mathe-matical programming algorithms and computer hardv^are make it possible to solve optimization problems of this scope for the first time. Delta is the first airline to solve to completion one of the largest and most difficult problems in this industry. Use of the Coldstart model is expected to save Delta Air Lines $300 million over the next three years. D elta Air Lines has over 2,500 domes- It has been said that an airline seat is the tic flight departures every day. This most perishable commodity in the world, includes flights to Canada and Mexico but Each time an airliner takes off with an excludes the other international routes. empty seat, a revenue opportunity is lost Delta has about 450 aircraft available to fly forever. So th...

Interior-point algorithms are nowadays among the most efficient techniques for processing monotone complementarity problems. In this paper, a procedure for globalizing interiorpoint methods by using the maximum stepsize is introduced. The algorithm combines exact or inexact interior-point and projected-gradient search techniques and employs a line-search procedure for the natural merit function associated to the complementarity problem. Furthermore for linear complementarity problems, the maximum stepsize is shown to be accepted in all iterations employing the exact interior-point search direction. A number of classes of complementarity and optimization problems are discussed, which the algorithm is able to process by either finding a solution or showing that no solution exists. A modification of the algorithm for dealing with infeasible linear complementarity problems is introduced, which solely employs interior-point search directions in practice. Computational experience on the solution of complementarity problems and linear and convex quadratic programs by the new algorithm is included, which illustrates the efficiency of this methodology.

The properties of the barrier F (x) = -log(det(x)), defined over the cone of squares of a Euclidean Jordan algebra, are analyzed using pure algebraic techniques. Furthermore, relating the Carathéodory number of a symmetric cone with the rank of an underlying Euclidean Jordan algebra, conclusions about the optimal parameter of F are suitably obtained. Namely, in a more direct and suitable way than the one presented by Osman Güler and Levent Tunçel (Characterization of the barrier parameter of homogeneous convex cones, Mathematical Programming, 81 (1998): 55-76), it is proved that the Carathéodory number of the cone of squares of a Euclidean Jordan algebra is equal to the rank of the algebra. Then, taking into account the result obtained in the same paper where it is stated that the Carathéodory number of a symmetric cone Q is the optimal parameter of a self-concordant barrier defined over Q, we may conclude that the rank of every underlying Euclidean Jordan algebra is also the self-concordant barrier optimal parameter.

It is well known that the Schatten p-norm defined on the space of matrices is useful and possesses nice properties. In this paper, we explore the concept of Schatten p-norm on R via the structure of Euclidean Jordan algebra. Two types of Schatten p-norm on R are defined and the relationship between these two norms is also investigated.

We introduce an entropy-like proximal algorithm for the problem of minimizing a closed proper convex function subject to the symmetric cone constraint. The algorithm is based on a distance-like function that is an extension of the Kullback-Leiber relative entropy to the setting of symmetric cones. Like the proximal algorithm for convex programming with nonnegative orthant cone constraint, we show that, under some mild assumptions, the sequence generated by the proposed algorithm is bounded and every accumulation point is a solution of the considered problem. In addition, we also present a dual application of the proposed algorithm to the symmetric cone linear program, leading to a multiplier method which is shown to enjoy properties similar to the exponential multiplier method in [29].

Based on the techniques of Euclidean Jordan algebras, we prove complexity estimates for a long-step primal-dual interior-point algorithm for the optimization problem of the minimization of a linear function on a feasible set obtained as the intersection of an a ne subspace and a symmetric cone. This result provides a meaningful illustration of a power of the technique of Euclidean Jordan algebras applied to problems under consideration.

In this paper we consider the question of solving equilibrium problems-formulated as complementarity problems and, more generally, mathematical programs with equilibrium constraints (MPECs)-as nonlinear programs, using an interior-point approach. These problems pose theoretical difficulties for nonlinear solvers, including interior-point methods. We examine the use of penalty methods to get around these difficulties and provide substantial numerical results. We go on to show that penalty methods can resolve some problems that interior-point algorithms encounter in general.

The Karush-Kuhn-Tucker (KKT) equations provide both necessary and sufficient conditions for optimality of convex linearly-constrained quadratic programming problems. These equations consist of both linear equations (the primal and dual feasibility constraints) and nonlinear equations (the complementary slackness conditions, specifying that the product of each complementary pair of variables is zero). While for many years complementary pivoting algorithms to solve the KKT equations were the state of the art for quadratic programming problems, more recently there has been much progress in the development of interior-point methods. Path-following interior-point algorithms proceed by relaxing the complementarity constraints of the KKT equations, specifying that each complementary product equals a positive parameter µ , which is successively reduced at each iteration. The solutions of the relaxed KKT equations constitute the "central path", parameterized by µ , converging to the optimum as µ approaches zero. For each µ , these equations are typically "solved" by one iteration of Newton's method, in which the nonlinear (complementarity) equations are approximated by linear equations and the resulting system of linear equations are then solved. The resulting sequence of points satisfy the primal and dual feasibility constraints exactly, but the (relaxed) complementarity conditions only approximately. An alternate approach, the so-called "monomial method", treats the complementarity constraints directly, and approximates the linear equations by monomial equations in order to obtain a system of equations which, after a logarithmic transformation, yields a linear system of equations. The sequence of points which results from solving these log-linear equations therefore satisfy the (relaxed) complementarity equations exactly, but the primal and dual feasibility constraints only approximately. This paper discusses this new path-following algorithm for quadratic programming, and evaluates its performance by presenting the results of some numerical experiments.

Due to the ability of modeling multivariable systems and handling constraints in the control framework, model predictive control (MPC) has received a lot of interest from both academic and industrial communities. Although it is an established control technique, implementing MPC on small-scale devices is a challenge since we need to handle complicated issues of the control framework using limited computational power and hardware resources. This paper presents our implementation of MPC with constraints on the Texas Instruments MSP430 16-bit microcontroller platform. The MPC operational constraints which are supported in our design include rate of change, amplitude and output constraints, while the associated optimization problem is solved using a primal-dual interior-point algorithm based on predicator-corrector method. Our implementation is demonstrated in a prototype of a real-time closeloop blood glucose regulation system using a modification of the minimal model. Experimental results show that our system is able to achieve desired diabetes management, and the chosen microprocessor is capable of performing the MPC algorithm accurately with high energy-efficiency and in realtime.

The use of three-phase voltage inverters (DC to AC converters) is frequently met in the electric power system, such as in the connection of photovoltaics with the rest of the grid. The paper proposes a nonlinear feedback control method for three-phase inverters, which is based on differential flatness theory and a new nonlinear filtering method under the name Derivative-free nonlinear Kalman Filter. First, it is shown that the inverter's dynamic model is a differentially flat one. This means that all its state variables and the control inputs can be written as functions of a single algebraic variable which is the flat output. By exploiting differential flatness properties it is shown that the inverter's model can be transformed to the linear canonical (Brunovsky's) form. For the latter description the design of a state feedback controller becomes possible, e.g. using pole placement methods. Moreover, to estimate the non-measurable state variables of the linearized equivalent of the inverter, the Derivative-free nonlinear Kalman Filter is used. This consists of the Kalman Filter recursion applied on the linearized inverter's model and of an inverse transformation that is based on differential B G. Rigatos

The interior point method is applied to frictionless contact mechanics problems and is shown to be a viable alternative to the augmented Lagrangian approach. The method is derived from a mixed formulation which induces a contact discretization scheme in the spirit of the mortar method and naturally delivers slack variables that help constrain the solution to the feasible region. The derivation of the algorithm as well as its robustness benefits from the contact interface description that is induced by NURBS-based isogeometric volume discretizations. Various interior point algorithms are discussed, including a primal-dual approach that satisfies the unilateral contact constraints exactly, in addition to two primal approaches that retain an arbitrary barrier parameter. The developed algorithms can easily be pursued starting from an augmented Lagrangian implementation. Numerical investigations on benchmark problems demonstrate the efficiency and the robustness of the framework, but also highlight current limitations that suggest paths for future research. Overall, the results indicate that the interior point method can challenge the augmented Lagrangian method in contact mechanics, even displaying potential for higher efficiency and robustness.

Sparse LU factorization offers some potential for parallelism, but at a level of very fine granularity. However, most current distributed memory MIMD architectures have too high communication latencies for exploiting all parallelism available. To cope with this, latencies must be avoided by coarsening the granularity and by message fusion. However, both techniques limit the concurrency, thereby reducing the scalability. In this paper, an implementation of a parallel LU decomposition algorithm for linear programming bases is presented for distributed memory parallel computers with noticable communication latencies. Several design decisions due to latencies, including data distribution and load balancing techniques, are discussed. An approximate performance model is set up for the algorithm, which allows to quantify the impact of latencies on its performance. Finally, experimental results for an Intel iPSC/860 parallel computer are reported and discussed.

This paper presents the optimal control policy of an industrial low-density polyethylene (LDPE) plant. Based on a dynamic model of the whole plant, optimal feed profiles are determined to minimize the transient states generated during the switching between different steady states. The industrial process under study produces LDPE by high-pressure polymerization of ethylene in a tubular reactor using oxygen and organic peroxides as initiators. The plant produces polyethylene of different grades that require continuous changes from one steady state to another, in order to switch among the different final products. These changes generate disturbances that keep the product out of specifications during the transient states, with a consequent economic loss. The plant model consists of two parts; the first one corresponds to the tubular reactor. Here, the partial differential equations corresponding to the mass and energy dynamic balances are discretized along the distance coordinate by using finite differences. The resulting ordinary differential equations include the energy balance and individual mass balances for oxygen, peroxides, ethylene, butane, free radicals and polymer. Although, methane is also present in the plant, in the reactor model it is considered as a nonreacting impurity along with the other impurities coming from the rest of the process. The second part of the model corresponds to the rest of the plant. Here we considered four components: ethylene, butane, methane and impurities. An interesting aspect of this process is the presence of many time delays that are incorporated in the optimization model. The resulting differential algebraic equation (DAE) plant model includes over five hundred equations. The dynamic optimization problem is solved using a simultaneous nonlinear programming (NLP) approach. The continuous state and control variables are discretized, by applying orthogonal collocation on finite elements. The resulting NLP is solved with a reduced space Interior Point Algorithm, which is applied directly to the NLP. In addition, a new mesh refinement strategy is applied to this model to confirm that no further improvement can be found in the optimal control profiles. The paper studies two cases of switching among different polymer grades, determining the optimal profiles of butane fed to the plant, in order to minimize the time to reach the steady state operation corresponding to the desired new product quality. The results are also compared with simpler model where reactor was considered as a black-box with the conversion level taken as constant data for each polymer grade. As a result, the dynamic model we developed and the solution methodology used is a flexible and practical tool to help process engineers for taking decisions during the plant operation.

As a natural extension of Roos and Vial's "Long steps with logarithmic penalty barrier function in llnear programming" (1989) and Ye's "An O(n 3 L) potential reduction algorithm for linear programming" (1989), it will be shown that the classical logarithmic barrier function method can be adjusted so that it generates the optimal solution in O(/IlL) iterations, where n is the number of variables and L is the data length.

Selforganizology (ISSN 2410-0080)
E-mail: [email protected]

Volume 12, Number 1-2, 1 June 2025
http://www.iaees.org/publications/journals/selforganizology/articles/2025-12(1-2)/2025-12(1-2).asp

Article

Using the analytic hierarchy process in an interactive interior point algorithm for mathematical multiple-objective nonlinear programming problems
Selforganizology, 2025, 12(1-2): 1-20
M. Tlas
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An interactive interior point method for solving multiple-objective nonlinear programming problems has been proposed. The method uses a single-objective nonlinear variant based on both logarithmic barrier function and Newton's method in order to generate, at each iterate, interior search directions. New feasible points are found along these directions which will be later used for deriving best-approximation to the gradient of the implicitly-known utility function at the current iterate. Using this approximate gradient, a single feasible interior direction for the implicitly-utility function could be found by solving a set of linear equations. It may be easily taken an interior step from the current iterate to the next one along this feasible direction. During the execution of the algorithm, a sequence of interior points will be generated. It has been proved that this sequence converges to an e-optimal solution, where e is a predetermined error tolerance known a priori. A numerical multiobjective example is illustrated using this algorithm.

http://www.iaees.org/publications/journals/selforganizology/articles/2025-12(1-2)/1-Tlas-Abstract.asp

The design and implementation of a new algorithm for solving large nonlinear programming problems is described. It follows a barrier approach that employs sequential quadratic programming and trust regions to solve the subproblems occurring in the iteration. Both primal and primal-dual versions of the algorithm are developed, and their performance is illustrated in a set of numerical tests.

We present a new strategy for choosing primal and dual steplengths in a primal-dual interior-point algorithm for convex quadratic programming. Current implementations often scale steps equally to avoid increases in dual infeasibility between iterations. We propose that this method can be too conservative, while safeguarding an unequally-scaled steplength approach will often require fewer steps toward to a solution. Computational results are given.

The k-means problem is one of the most popular models in cluster analysis that minimizes the sum of the squared distances from clustered objects to the sought cluster centers (centroids). The simplicity of its algorithmic implementation encourages researchers to apply it in a variety of engineering and scientific branches. Nevertheless, the problem is proven to be NP-hard which makes exact algorithms inapplicable for large scale problems, and the simplest and most popular algorithms result in very poor values of the squared distances sum. If a problem must be solved within a limited time with the maximum accuracy, which would be difficult to improve using known methods without increasing computational costs, the variable neighborhood search (VNS) algorithms, which search in randomized neighborhoods formed by the application of greedy agglomerative procedures, are competitive. In this article, we investigate the influence of the most important parameter of such neighborhoods on the c...

We propose a new technique for minimization of convex functions not necessarily smooth. Our approach employs an equivalent constrained optimization problem and approximated linear programs obtained with cutting planes. At each iteration a search direction and a step length are computed. If the step length is considered "non serious", a cutting plane is added and a new search direction is computed. This procedure is repeated until a "serious" step is obtained. When this happens, the search direction is a feasible descent direction of the constrained equivalent problem. The search directions are computed with FDIPA, the Feasible Directions Interior Point Algorithm. We prove global convergence and solve several test problems very efficiently.

A mathematical model for the computation of the phase equilibrium related to atmospheric organic aerosols is presented. The phase equilibrium is given by the global minimum of the Gibbs free energy for a system that involves water and organic components. This minimization problem is equivalent to the determination of the convex hull of the corresponding molar Gibbs free energy function. A geometrical notion of phase simplex related to the convex hull is introduced to characterize mathematically the phases at equilibrium. A primal-dual interior-point algorithm for the efficient solution of the phase equilibrium problem is presented. A novel initialization of the algorithm, based on the properties of phase simplex, is proposed to ensure the convergence to a global minimum of the Gibbs free energy. For a finite termination of the interior-point method, an active phase identification procedure is incorporated. Numerical results show the robustness and efficiency of the approach for the prediction of liquid-liquid equilibrium in multicomponent mixtures.

A mathematical model for the computation of the phase equilibrium and gas-particle partitioning in atmospheric organic aerosols is presented. The thermodynamic equilibrium is determined by the global minimum of the Gibbs free energy under equality and inequality constraints for a system that involves one gas phase and many liquid phases. A primal-dual interior-point algorithm is presented for the efficient solution of the phase equilibrium problem and the determination of the active constraints. The first order optimality conditions are solved with a Newton iteration. Sequential quadratic programming techniques are incorporated to decouple the different scales of the problem. Decomposition methods that control the inertia of the matrices arising in the resolution of the Newton system are proposed. A least-squares initialization of the algorithm is proposed to favor the convergence to a global minimum of the Gibbs free energy. Numerical results show the efficiency of the approach for the prediction of gas-liquid-liquid equilibrium for atmospheric organic aerosol particles.

1. Abstract In the classical approach for Engineering Design Optimization, a Mathematical Program that depends on the design variables is solved. That is, the objective function and the constraints depend exclusively of the design variables. Thus, the state equation that ...

The Interior Point Methods primal-dual when applied to problems of optimal power flow has great results, but when the system presents overloads in generation and/or transmission, not converge because these overloads imply violation of operating limits, causing some variables cease to be interiors. In order to eliminate these difficulties of these methods is proposed an exchange of barrier function, replacing the classical logarithmic barrier by modified logarithmic barrier. This change allows for controlled violations in some inequality constraints and can be used in solving problems such as optimal power flow with overloads. In practice, such violations can be performed in a short period without damaging the system. Finally, computational tests are made simulating overloads the system suggesting good results.

The Interior Point Methods primal-dual when applied to problems of optimal power flow has great results, but when the system presents overloads in generation and/or transmission, not converge because these overloads imply violation of operating limits, causing some variables cease to be interiors. In order to eliminate these difficulties of these methods is proposed an exchange of barrier function, replacing the classical logarithmic barrier by modified logarithmic barrier. This change allows for controlled violations in some inequality constraints and can be used in solving problems such as optimal power flow with overloads. In practice, such violations can be performed in a short period without damaging the system. Finally, computational tests are made simulating overloads the system suggesting good results.

Interior-point algorithms are nowadays among the most efficient techniques for processing monotone complementarity problems. In this paper, a procedure for globalizing interiorpoint methods by using the maximum stepsize is introduced. The algorithm combines exact or inexact interior-point and projected-gradient search techniques and employs a line-search procedure for the natural merit function associated to the complementarity problem. Furthermore for linear complementarity problems, the maximum stepsize is shown to be accepted in all iterations employing the exact interior-point search direction. A number of classes of complementarity and optimization problems are discussed, which the algorithm is able to process by either finding a solution or showing that no solution exists. A modification of the algorithm for dealing with infeasible linear complementarity problems is introduced, which solely employs interior-point search directions in practice. Computational experience on the solution of complementarity problems and linear and convex quadratic programs by the new algorithm is included, which illustrates the efficiency of this methodology.

We investigate the relation between interior-point algorithms applied to two homogeneous self-dual approaches to linear programming, one of which was proposed by Ye, Todd, and Mizuno and the other by Nesterov, Todd, and Ye. We obtain only a partial equivalence of path-following methods (the centering parameter for the first approach needs to be equal to zero or larger than one half), whereas complete equivalence of potential-reduction methods can be shown. The results extend to self-scaled conic programming and to semidefinite programming using the usual search directions.

In this paper we continue the development of a theoretical foundation for efficient primal-dual interior-point algorithms for convex programming problems expressed in conic form, when the cone and its associated barrier are self-scaled (see [NT97]). The class of problems under consideration includes linear programming, semidefinite programming and convex quadratically constrained quadratic programming problems. For such problems we introduce a new definition of affine-scaling and centering directions. We present efficiency estimates for several symmetric primal-dual methods that can loosely be classified as path-following methods. Because of the special properties of these cones and barriers, two of our algorithms can take steps that go typically a large fraction of the way to the boundary of the feasible region, rather than being confined to a ball of unit radius in the local norm defined by the Hessian of the barrier.

In this paper we study strong duality aspects in convex conic programming over general convex cones. It is known that the duality in convex optimization is linked with specific theorems of alternatives. We formulate and prove strong alternatives to the existence of the relative interior point in the primal (dual) feasible set. We analyze the relation between the boundedness of the optimal solution sets and the existence of the relative interior points in the feasible set. We also provide conditions under which the duality gap is zero and the optimal solution sets are unbounded. As a consequence, we obtain several alternative conditions that guarantee the strong duality between primal and dual convex conic programs.

A new algorithm is presented for size optimization of truss structures with any kind of smooth objectives and constraints, together with constraints on the collapse loading obtained by limit analysis, for several loading conditions. The main difficulty of this problem is the fact that the collapse loading is a nonsmooth function of the design variables. In this paper we avoid nonsmooth optimization techniques based on the fact that limit analysis constraints are linear by parts. Our approach is based on a Feasible Directions Interior Point Algorithm for nonlinear constrained optimization. Three illustrative examples are discussed. The numerical results show that the calculation effort when limit analysis constraints are included is only slightly increased with respect to classic constraints.

Mathematical Programming provides general tools for Engineering Design Optimization. We present numerical models for Simultaneous Analysis and Design Optimization (SAND) and Multidisciplinary Design Optimization (MDO) represented by Mathematical Programs and numerical techniques to solve these models. These techniques are based on the Feasible Arc Interior Point Algorithm (FAIPA) for Nonlinear Constrained Optimization. Even if MDO is a very large optimization problem, our approach reduces considerably the computer effort. Several tools for very large problems are also presented. The present approach is very strong and efficient for real industrial applications and can easily interact with existing simulation engineering codes.

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro Tecnológico.O presente trabalho tem como objetivo apresentar o desenvolvimento de um procedimento flexível e de baixo custo, que auxilie o processo de otimização estrutural em um ambiente de projeto. Para isto foi tomado como premissa o fato de que em muitos ambientes de projeto é comum encontrar uma infinidade de ferramentas computacionais tipo CAD, CAM e CAE. Com relação as ferramentas de CAE, normalmente, estas são pacotes básicos, limitados à análises simples, como pôr exemplo, estática linear, térmica sem regime transiente e modal, ou ainda programas sofisticados, contudo fechados. Neste sentido, desenvolveu-se um procedimento para otimização de projeto estrutural, integrando a tecnologia existente com algoritmos de otimização. Em linhas gerais, o procedimento foi desenvolvido a partir da base de elementos finitos já instalada, ou seja, utilizando-se de um programa comercial, como o programa comercial Ansys...

A full Nesterov-Todd (NT) step infeasible interior-point algorithm is proposed for solving monotone linear complementarity problems over symmetric cones by using Euclidean Jordan algebra. Two types of full NT-steps are used, feasibility steps and centering steps. The algorithm starts from strictly feasible iterates of a perturbed problem, and, using the central path and feasibility steps, finds strictly feasible iterates for the next perturbed problem. By using centering steps for the new perturbed problem, strictly feasible iterates are obtained to be close enough to the central path of the new perturbed problem. The starting point depends on two positive numbers ρp and ρ d. The algorithm terminates either by finding an ϵ-solution or detecting that the symmetric cone linear complementarity problem has no optimal solution with vanishing duality gap satisfying a condition in terms of ρp and ρ d. The iteration bound coincides with the best known bound for infeasible interior-point methods.

This paper describes an interactive approach for multiobjective linear programming problems combining the tolerance approach to sensitivity analysis with the reference point methodology. It aims at providing decision makers with a flexible decision aid tool, enabling to perform a progressive search of the set of efficient solutions, and which does not require too much computational effort. The proposed combined approach enables to gather, in an interactive manner, information about the problem and the trade-offs to be made among the objectives in order to obtain a compromise solution, which a final decision can be based on. The main concepts will be geometrically illustrated with a simple example.

We propose an algorithm for the effective solution of quadratic programming (QP) problems arising from model predictive control (MPC). MPC is a modern multivariable control method which gives the solution for a QP problem at each sample instant. Our algorithm combines the active-set strategy with the proportioning test to decide when to leave the actual active set. For the minimization in the face, we use a direct solver implemented by the Cholesky factors updates. The performance of the algorithm is illustrated by numerical experiments, and the results are compared with the state-of-the-art solvers on benchmarks from MPC.

This note outlines an algorithm for solving the complex "matrix Procrustes problem". This is a least-squares approximation over the cone of positive semi-definite Hermitian matrices, which has a number of applications in the areas of Optimization, Signal Processing and Control. The work generalises the method of [1], who obtained a numerical solution to the realvalued version of the problem. It is shown that, subject to an appropriate rank assumption, the complex problem can be formulated in a real setting using a matrix dilation technique, for which the method of [1] is applicable. However, this transformation results in an over-parametrisation of the problem and, therefore, convergence to the optimal solution is slow. Here an alternative algorithm is developed for solving the complex problem, which exploits fully the special structure of the dilated matrix. The advantages of the modified algorithm are demonstrated via a numerical example. In addition the following notation is used: The range and nullspace of a matrix A are written as Range(A) and Null(A), respectively. The convex hull of a nonempty set X is denoted by conv[X] and the conical hull of X, i.e. the set conv[{αx : α ≥ 0, x ∈ X}] by cone[X]. If A ∈ C m×n and X ⊆ C n then AX denotes the set {Ax : x ∈ X} ⊆ C m. The line {αx + (1 − α)y : α ∈ [0, 1]} between two points x and y is written as line{x, y}. The unique point x in a closed, convex set X which minimises d − x with respect to x ∈ X is denoted by minpoint[d, X] and the corresponding minimal distance d −x by mindist[d, X].

Bilinear inverse problems (BIPs), the resolution of two vectors given their image under a bilinear mapping, arise in many applications. Without further constraints, BIPs are usually ill-posed. In practice, properties of natural signals are exploited to solve BIPs. For example, subspace constraints or sparsity constraints are imposed to reduce the search space. These approaches have shown some success in practice. However, there are few results on uniqueness in BIPs. For most BIPs, the fundamental question of under what condition the problem admits a unique solution, is yet to be answered. For example, blind gain and phase calibration (BGPC) is a structured bilinear inverse problem, which arises in many applications, including inverse rendering in computational relighting (albedo estimation with unknown lighting), blind phase and gain calibration in sensor array processing, and multichannel blind deconvolution (MBD). It is interesting to study the uniqueness of such problems. In this paper, we define identifiability of a BIP up to a group of transformations. We derive necessary and sufficient conditions for such identifiability, i.e., the conditions under which the solutions can be uniquely determined up to the transformation group. These conditions take the form of dividing the identifiability of the pair of unknown variables into the individual identifiability of each variable. Although verifying these individual conditions requires problem-specific procedures, this framework is universally applicable to all BIPs. Applying these results to BGPC, we derive sufficient conditions for unique recovery under several scenarios, including subspace, joint sparsity, and sparsity models. For BGPC with joint sparsity or sparsity constraints, we develop a procedure to compute the transformation groups corresponding to inherent ambiguities. We also give necessary conditions in the form of tight lower bounds on sample complexities, and demonstrate the tightness of these bounds by numerical experiments. The results for BGPC not only demonstrate the application of the proposed general framework for identifiability analysis, but are also of interest in their own right.

Expansion algorithm is a popular optimization method for labeling problems. For many common energies, each expansion step can be optimally solved with a min-cut/max flow algorithm. While the observed performance of max-flow for the expansion algorithm is fast, its theoretical time complexity is worse than linear in the number of pixels. Recently, Dynamic Programming (DP) was shown to be useful for 2D labeling problems via a "tiered labeling" algorithm, although the structure of allowed (tiered) is quite restrictive. We show another use of DP in a 2D labeling case. Namely, we use DP for an approximate expansion step. Our expansion-like moves are more limited in the structure than the max-flow expansion moves. In fact, our moves are more restrictive than the tiered labeling structure, but their complexity is linear in the number of pixels, making them extremely efficient in practice. We illustrate the performance of our DP-expansion on the Potts energy, but our algorithm can be used for any pairwise energies. We achieve better efficiency with almost the same energy compared to the max-flow expansion moves.

For the iterative solution of the matrix equation A x = b by means of the (point) symmetrie SOR method (called the SSOR method), the basic convergence analysis of this iterative process has been developed in the literature only for the case when A is Hermitian and positive definite. With the help of the theory of regular splittings, a more general convergence analysis of this iterative method is obtained, under the weaker assumption th2t A is a nonsingular H-matrix.

In [3] hat R. S. Varga den Begriff der regulären Zerlegung einer Matrix eingeführt und damit die Konvergenz einiger Iterationsverfahren zur Auflösung linearer Gleichungssysteme nachgewiesen. In dieser Arbeit sollen diese Ergebnisse in zweierlei Hinsicht verallgemeinert werden: Einerseits werden auch nichtlineare Abbildungen zugelassen, andererseits werden alle Aussagen in einem halbgeordneten Banachraum formuliert und bewiesen. Als konkrete Anwendung ergeben sich unmittelbar Aussagen über die Konvergenz einiger Iterationsverfahren zur Auflösung nichtlinearer Gleichungssysteme im Rn.

Ha muito tempo atras, por inumeros motivos, e necessario resolver sistemas de equacoes. Existem varios metodos numericos iterativos que aproveitam a ajuda dos computadores para este proposito, dos quais podemos destacar dois: o metodo de Newton e o metodo de Brent. Ambos algoritmos podem ser aplicados sempre que F , a funcao vetorial descrita pelas mesmas equacoes do sistema, seja de classe C 1 . No primeiro metodo, em cada iteracao sao consideradas todas as equacoes do problema simultaneamente; no segundo, as equacoes sao particionadas em blocos disjuntos e em cada iteracao cada bloco e considerado separadamente e de forma ordenada, tentando manter o progresso obtido pelo tratamento dos blocos anteriores. Ambos metodos calculam a solucao de sistemas lineares consistentes em apenas uma iteracao, como e demonstrado em [1]. No caso de sistemas nao lineares consistentes, para ambos metodos e possivel mostrar resultados de convergencia local quadratica, com algumas hipoteses adicionais ...

Nesse trabalho serão propostos métodos de Lagrangiano Aumentado para tratar problemas com restrições do tipo LOVO, serão propostos novos métodos de Restauração Inexata e será introduzido o conceito de Equilíbrio Inverso de Nash. Teoremas sobre condições de otimalidade para problemas do tipo LOVO serão apresentados. Um algoritmo do tipo Lagrangiano Aumentado será proposto para abordar esse problema e teoremas de convergência global serão demonstrados. Resultados computacionais serão realizados para uma aplicação em otimização de carteiras em investimentos de grande impacto. Um método híbrido de Restauração Inexata será proposto combinando uma modificação, que usa o Lagrangiano Afiado como função de mérito, do método global de Fischer e Friedlander e o método local de Birgin e Martínez. Teoremas de convergência global e local serão apresentados. Um método de Restauração Inexata para problemas em que as derivadas da função objetivo não estejam disponíveis será introduzido. Nesse método todas as ferramentas da otimização tradicional serão usadas na fase de restauração e uma regularização será feita na fase de otimização. Teoremas de convergência global serão demonstrados e resultados numéricos apresentados. O conceito de Equilíbrio Inverso de Nash será introduzido e um método de Restauração Inexata será proposto para abordar esse problema. Esse método será uma extensão de um novo método de Restauração Inexata para problemas em dois níveis que também será proposto neste trabalho. Exemplos ilustrativos para uma aplicação para o problema de equilíbrio de Arrow-Debreu serão exibidos.

Primal-dual Interior-Point Methods (IPMs) have shown their ability in solving large classes of optimization problems efficiently. Feasible IPMs require a strictly feasible starting point to generate the iterates that converge to an optimal solution. The self-dual embedding model provides an elegant solution to this problem with the cost of slightly increasing the size of the problem. On the other hand, Infeasible Interior Point Methods (IIPMs) can be initiated by any positive vector, and thus are popular in IPM softwares. In this paper we propose an adaptive large-update IIPM based on a specific self-regular proximity function, with barrier degree 1 + log n, that operates in the infinity neighborhood of the central path. An O n 3 2 log n log n worst-case iteration bound of our new algorithm is established. This iteration bound improves the so far best O n 2 log n iterations bound of IIPMs in a large neighborhood of the central path.

The long-term planning of electricity generation in a liberalised market using the Bloom and Gallant model can be posed as a quadratic programming (QP) problem with an exponential number of linear inequality constraints called load-matching constraints (LMCs) and several other linear non-LMCs. Direct solution methods are inefficient at handling such problems and a heuristic procedure has been devised to generate only those LMCs that are likely to be active at the optimiser. The problem is then solved as a finite succession of QP problems with an increasing, though still limited, number of LMCs, which can be solved efficiently using a direct method, as would be the case with a QP interior-point algorithm. Warm starting between successive QP solutions helps then in reducing the number of iterations necessary to reach the optimiser. The warm start technique employed herein is an extension of Gondzio and Grothey's approach to quadratic programming problems. We also propose how to initialise new variables in the problem to which a warm start technique is applied. This study shows that warm starting requires on average 50% fewer iterations than a cold start in the test cases solved. The reduction in computation time is smaller, however.